Error analysis is an instructional strategy that can assist teachers to identify learners’ areas of weakness in mathematics and that can point to remediation of those errors. This article explores the errors learners exhibit when solving quadratic equations by completing the square using Newman’s Error Analysis Model. A research study explored the errors learners exhibit when solving quadratic equations by completing the square. Newman’s Error Analysis Model provided the analytic framework for the qualitative approach that was used to explore those errors. A diagnostic test with five test items was administered to 35 learners in one secondary school in Limpopo province of South Africa. Subsequently, 12 learners whose scripts featured common mistakes were identified; these learners participated in a semi-structured interview to diagnose the errors. The findings revealed that learners commit comprehension, transformation and process errors. The findings suggest that if the errors that learners make are exposed and made explicit, the errors can be remediated and thereby enhance understanding and learning. The findings of this study indicate that for teachers to understand the types of errors learners commit when solving quadratic equations by completing the square it is vital for them (errors) to be addressed. Mathematics teachers should also consider diagnosing why learners commit those errors, as they would know the strategies to be employed to teach this topic and subsequent topics.

The findings of this article add value to the current literature by providing empirical knowledge on learner challenges when solving quadratic equations by completing the square. This study provides opportunities for mathematics teachers to focus more on the strategies that would assist learners to understand this topic.

A quadratic equation (QE) is an algebraic equation of the second degree with one variable (Harripersaud, ^{2} +

Quadratic equations are a branch of mathematics that cut across all spheres and that need to be taught and learned in secondary schools (Cahyani & Rahaju,

According to Sari and Jailani (

Finzer and Bennett (^{2} + ^{2} be 1, when

To the best of the researcher’s knowledge, solving QE by CS with Grade 11 learners using Newman’s Error Analysis Model (EAM) is rarely investigated (Newman, ^{2} +

This article sought to explore the errors learners exhibited in relation to Newman’s EAM when solving QE by CS in Grade 11 mathematics. Using Newman’s EAM as a lens, the focus was on the comprehension error type, transformation error type and processing error type. This lens enabled the researcher to understand how learners explain and solve QE by CS, as well as their attributions of the sources of those errors. Methods of inquiry included semi-structured interviews and the outcomes of a diagnostic test to identify and diagnose errors learners exhibit in solving QE by CS, to make teachers aware of the pattern of the errors learners display to know the strategies teachers can employ when teaching this topic. The objectives of the article are to:

Identify the errors learners exhibited when solving QE by CS using Newman’s EAM.

Diagnose the reasons why learners exhibited those errors.

The research questions were, therefore: (1)

This article argues that Grade 11 learners have challenges in solving QE by CS and many exhibit errors that should have been addressed in Grade 10. The learners were found to have comprehension, transformation and processing errors when solving QE by CS. This study revealed learners’ lack of prior conceptual knowledge that could have taken the form of introduction to QE in Grades 10 and 11.

This article begins with a brief discussion of the SA curriculum orientation towards QE, errors in QE and the difficulties learners experience in solving QE, as found in the scholarly literature. Newman’s EAM is also discussed as a framework underpinning this study and explains the research methodology espoused to collect and analyse data that answer the research questions. Finally, strategies to be used by teachers, curriculum developers, mathematics specialists and textbook writers to address the errors and their cause in solving QE by CS are recommended.

Quadratic equations in the SA curriculum context is introduced in Grade 10 and both Grade 10 and Grade 11 learners should start solving the equations using factorisation (Curriculum and Assessment Policy Statement [CAPS],

Studies have demonstrated that most learners struggle with QE (Kim How et al.,

Makgakga (

Some types of errors identified are cognitive, as revealed in Díaz et al.’s (

Errors in QE committed by learners were also investigated by Abubaker (

Errors exhibited by learners when solving QE can recur and as a result affect learners’ learning of subsequent concepts (Sari & Jailani,

Newman’s (

The EAM has gained popularity in mathematics education on error analysis and has proven to be reliable in classifying and categorising learners’ errors. The approach is also used by Clarkson (

As earlier noted, successive diagnostic errors identified by Newman when solving mathematical problems are reading, comprehensive, transformative, processing and encoding. These errors are outlined as follows:

The reading stage examines learners’ ability or inability to read the statements to identify the key elements or main points relevant to the question to prepare for the next stage.

The second error type is the comprehensive stage that determines the learners’ inability or ability to comprehend the mathematical statements, break the problem into smaller chunks and make sense of it.

The third error type is the transformation stage which determines the ability or inability to choose mathematical operations or methods, and correct or incorrect procedures to solve mathematical problems.

The fourth error type is the processing stage in which learners execute mathematical procedures correctly or incorrectly.

Lastly, the encoding error type is where the learner can write the correct or incorrect answer but cannot justify the answer or provide the conclusion of the given answer.

Error analysis is important for teachers and researchers as it helps them to choose the appropriate approaches, strategies, instructional media and models to alleviate learners’ errors in mathematics (Fitriani et al.,

This qualitative exploratory case study design explored the errors learners exhibited in solving QE by CS and the reasons why those learners exhibit those errors applying the EAM. An exploratory case study is a way to understand what is happening, ask questions, seek new insights and assess a phenomenon in a new light (Yin,

A diagnostic test was administered to 35 Grade 11 learners (19 female and 16 male) in one of the rural secondary schools in the Limpopo province of SA. Eight QE problems adapted from previous Grade 11 examination papers (

The structure of the design of the test instrument.

Items | Motivation for question | |
---|---|---|

1. | Describe a quadratic equation. | To understand how learners describe a quadratic equation. |

2. | Describe the methods of the completing the square method. | To understand how learners describe the completing the square method when solving quadratic equations. |

3. | Give five procedures for completing the square method. | Learners are asked to give the five features of the completing the square method for them to be able to solve quadratic equations using this method. |

4. | ^{2} – 2 |
Learners were assessed on the QE with the coefficient of ^{2} equal to 1 and constant term as –1. |

5. | 2^{2} – 2 |
Learners were assessed on the QE with the coefficient of ^{2} greater than 1 and constant term as –9. |

6. | −3^{2} + 2 |
Learners were assessed on the QE with the coefficient of ^{2} less than 0 and constant term as +2. |

7. | −2^{2} + 3 |
Learners were assessed on the QE with the coefficient of ^{2} less than 1, coefficient of |

8. | ^{2} + |
Learners were required to derive the formula using the QE given by completing the square. |

All learners’ scripts were gathered immediately after they were completed and marked on the same day. A day after marking the test scripts, the researcher conducted semi-structured interviews of 15 minutes with four male and six female learners, purposively selected according to the types of errors committed in their assessment scripts to understand why they committed those errors. At the time of data collection, learners had learned QE by factorisation, CS, and using the quadratic formula according to the departmental curriculum guidelines, termed a pace setter. The researcher used the EAM during the interviews to determine the errors learners made when solving QE by CS. The collected data were analysed and interpreted by classifying and identifying error types according to Newman’s (

The researcher sought permission from a Grade 11 mathematics teacher and learners to participate in this study. The principal and head of department of mathematics and science in the school were informed about the research. The role and participation of learners was explained prior the inception of the study. Privacy and confidentiality of the learners was protected before and after the study. Consent forms were signed by the learners who were under 18 years of age at the time of the study to confirm their participation.

This section describes how Newman’s EAM is used to analyse the data sets collected for this study. The main concepts of the framework are defined and the performance indicator is described. The main concepts are comprehension error, reading error, transformation error, processing error, and encoding error.

The analysis of the findings applying Newman’s (

Few learners described a QE as an equation that can be solved by factorisation, completing the square or using the quadratic formula. Abubaker (^{2} +

Learners’ samples in describing quadratic equations: (a) L2F (b) L5F, and (c) L3M.

Learners were not able to describe or define a quadratic equation in terms of concepts or mathematical ideas – they resorted to giving methods of solving the equation. This lack of conceptual understanding underlies some difficulties the learners experience in solving QE using either one of the methods. This QE topic cuts across all spheres and should be taught at secondary schools (Cahyani & Rahaju,

However, most of the learners showed no comprehension in describing a QE for QI2 and QI3. Lack of comprehension is visible when a learner cannot describe what CS is when solving QE and know the five features of CS for them to solve QE using this method. Alhassan and Agyei (

This evidence (

Learners’ samples in describing the completing the square method: (a) L1M (b) L6F, and (c) L11M.

In addition to being unable to describe CS when solving QE, the majority of the learners could not mention the five features of CS (Laridon et al., ^{2} if

Transformation errors were also revealed in the learners’ scripts as they appeared not to know the mathematical operations, correct procedures, or methods in solving mathematical problems (Newman,

Examples of transformation error type: (a) L8M and (b) L9F.

Methodological approach.

Concept | Definition | Performance indicator | |
---|---|---|---|

1. | Comprehension error type | Determines learners’ ability or inability to understand QE. | Learners need to define a quadratic equation and describe the completing the square method as a technique to transform QE to make the left-hand side a perfect square trinomial and to give the features of this method. |

2. | Transformation error type | Learners’ ability or inability to choose mathematical methods or operations, correct or incorrect procedures to solve QE: ^{2} + |
Learners can transform the equation by first finding the additive inverse of ^{2} which is ^{2} + |

3. | Processing error type | Learners execute mathematical procedures correctly or incorrectly to solve QE. | Learners solve the problem using procedures that are correct or incorrect to get the answer. Learners need to complete the square by adding the square of half the coefficient of |

Learners revealed transformation errors as they could not correctly interpret the three terms in the equation; they followed the incorrect procedures to change the equation. Predominant procedural challenges can lead learners to commit errors when solving mathematical problems (Díaz et al., ^{2} should be equal to 1 before they could add the square of half the coefficient of ^{2}, which is

In the interview sessions learners who were selected to explain how they answered the QE questions showed difficulties when solving QE by CS. They seem to have misunderstood the topic as they could not justify the procedures used to solve QE problems and three of them (1 male and 2 female) indicated that the strategies used to teach this topic were not easy to understand.

Total number of errors according to test items and Newman’s Error Analysis Model.

Item | Reading | Comprehension |
Transformation |
Processing skills |
Encoding |
Total |
|||||
---|---|---|---|---|---|---|---|---|---|---|---|

% | % | % | % | % | |||||||

QI1 | 0 | 9 | 25.7 | 0 | - | 0 | - | 0 | - | 15 | 42.9 |

QI2 | 0 | 25 | 71.4 | 0 | - | 0 | - | 0 | - | 25 | 71.4 |

QI3 | 0 | 23 | 65.7 | 0 | - | 0 | - | 0 | - | 23 | 65.7 |

QI4 | 0 | 0 | - | 17 | 48.6 | 13 | 37.1 | 3 | 8.6 | 22 | 62.9 |

QI5 | 0 | 0 | - | 19 | 54.3 | 14 | 40.0 | 1 | 2.9 | 26 | 74.3 |

QI6 | 0 | 0 | - | 18 | 51.4 | 13 | 37.1 | 4 | 11.4 | 25 | 71.4 |

QI7 | 0 | 0 | - | 16 | 45.7 | 15 | 42.9 | 2 | 5.7 | 28 | 80.0 |

QI8 | 0 | 0 | - | 15 | 42.9 | 17 | 48.6 | 3 | 8.6 | 45 | 25.7 |

L8M response on QI4 (Excerpt 1).

Speaker | Dialogue |
---|---|

Interviewer | ‘Okay, can you explain how did you arrive in step 2?’ [ |

L8M | [ |

Interviewer | [^{2} ^{2} in the equation?’ |

L8M | [ |

Interviewer | ‘Then what is |

L8M | ‘Yah, I see now [ |

Interviewer | ‘Okay, but why didn’t you solve the equation?’ |

L8M | ‘This is difficult for me. The teacher was also moving in a fast pace when teaching this topic.’ |

Interviewer | ‘Is there anything you would like [ |

L8M | ‘Yes, maybe the teacher to give us extra lesson and not to be fast in teaching the topic.’ |

The two excerpts (

L9F response on QI4 (Excerpt 2).

Speaker | Dialogue |
---|---|

Interviewer: | ‘Good, how do you understand this question?’ |

L9F | ‘Yah Sir, |

Interviewer | ‘Okay, what do you mean by completing a square?’ |

L9F | ‘We have to add half the square of the coefficient of |

Interviewer | ‘Yah [ |

L9F | ‘Yes, we are done and we can solve the equation.’ |

Interviewer | ‘Then, let’s look at the equation [^{2} –2^{2} in the equation? Does it have any meaning?’ |

L9F | ‘Okay [ |

Interviewer | [ |

L9F | ‘Eish, I made a mistake in my script, did not realise it when solving the equation. The answer is 1 because |

Interviewer | [ |

L9F | [ |

Interviewer | ‘Good, then why didn’t you finish solving the equation?’ |

L9F | ‘Eish, completing a square is challenging, I did not know what to do further. Even the way we are taught, it’s a problem as I did not understand it in Grade 10. The teacher was fast to cover the scope of the syllabus.’ |

Another reason why learners display transformation errors is that of multiplying both sides by the multiplicative inverse of 2, which is the coefficient of ^{2}, ^{2} where a ≠ 1 and finding the additive inverse of the constant term are the first two essential features that can assist learners to change the original equation to a new equation to complete a square. Here, the new equation was supposed to be

The learner showed little understanding of what has to be done to solve QI2 by CS. The transformation error type is identified as L9F did not know that the coefficient of ^{2} needs to be equal to 1 before adding the square of half the coefficient of

The other error type found in this study is, according to the EAM framework, a processing error. The processing error type is when learners follow incorrect procedures or mathematical operations or methods to execute the problem (Newman, ^{2} greater than or less than 1, errors occurred when learners could not make the coefficient of ^{2} equal to 1 which led them to use incorrect procedures to solve the equations.

L19F and L26M in their scripts (^{2} equal 1. Although L19F and L26M followed the wrong procedures to solve the equation, there is consistency in procedures used in the process to arrive at the answer.

Examples of processing error type (QI8): (a) L19F and (b) L26M.

^{2} + ^{2}, which is

Interview with L19F (Excerpt 3).

Speaker | Dialogue |
---|---|

Interviewer | ‘Okay, what does this need you to do?’ |

L19F | ‘This question wanted us to complete a square, but my challenge is that the equation has no numbers [^{2} be 1.’ |

Interviewer | ‘Great, then why did you have |

L19F | ‘Eish [ |

Interviewer | ‘Okay, then what about this |

L19F | ‘Okay Sir, as |

Interviewer | ‘Good thank you, then is that all with this [ |

L19F | ‘Yes Sir, we can remove the brackets and my answer is – |

Interviewer | ‘Do you want to say anything before we finish with our interview session?’ |

L19F | ‘Yes Sir, the teacher needs not to be fast when teaching this topic as it is challenging.’ |

Interview with L19F (Excerpt 4).

Speaker | Dialogue |
---|---|

Interviewer | ‘Okay, how can you solve this equation [^{2} + |

L26M | ‘Hmm [ |

Interviewer | ‘Alright then, [ |

L26M | ‘Yes, but I must find the additive inverse of |

Interviewer | ‘Okay, what can you say about the coefficient of ^{2}, |

L26M | ‘It is just that this topic is difficult for me [^{2} was used as a common factor [ |

Interviewer | ‘Why do you say the topic is difficult?’ |

L26M | ‘Most of us cannot solve quadratic equations by completing a square. The teacher moved in a fast pace when teaching this topic. It really frustrates us.’ |

Interviewer | ‘Is there anything you can share before we finish with our interview session?’ |

L26M | ‘I think the teacher can move in a slower pace when teaching this topic and other mathematical topics.’ |

This study intended to explore the errors learners exhibit when solving QE by CS. This study revealed a higher rate of comprehension, transformation, and processing skills. No reading error was identified, and a low rate of encoding error was found. Tendere and Mutambara (^{2} not equal to 1 (^{2} in QI5–QI8 to prepare for the transformation stage. This comprehension error type could be the result of the misapplication of the learned procedures caused by carelessness, slips or silly mistakes committed by learners (Tendere & Mutambara,

In the transformation error type, learners grappled with QE problems when completing the square on both sides, as most of them completed the square on one side; as such, the coefficient of ^{2} was not made 1. Similar findings are found by Mahmud et al. (^{2} – 2

The other error type was found at the processing stage where learners were expected to use procedures to determine the values of

No reading error was identified, and this is supported by Thomas and Mahmud’s (

This study explored the errors learners exhibit when solving QE by CS using Newman’s EAM. It argues that Grade 11 mathematics learners need to have a good background in QE by factorisation, CS and using the quadratic formula to avoid making errors when solving QE problems. The study revealed that learners committed comprehension errors, transformation errors and processing errors when solving QE by CS.

Firstly, the comprehension error type showed learners grappling to describe QE and CS. The study revealed learners misinterpreting and not mentioning the five key features of CS, especially with equations where the coefficient of ^{2} equal to 1 (

Secondly, the transformation error type included multiplicative errors, additive errors and incorrect choices of coefficients when preparing to solve the equations. Most learners treated the equations the same when applying CS, adding the square of half the coefficient of

Learners’ errors need the teachers’ intervention for learners to learn subsequent concepts (Díaz et al.,

The author acknowledge the participants of this study who wrote a diagnostic test and took part in the semi-structured interviews. The author also acknowledge Prof. Caroline Long who critically reviewed the manuscript and Ms Joyce Musi who contributed to language editing.

The author declare that he has no financial or personal relationships that may have inappropriately influenced him in writing this article.

T.P.M. is the sole author of this article.

An ethical clearance certificate was obtained from the College of Education Ethics Committee, reference number: 2022/02/09/90197607/14/AM.

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

The authors confirm that the data supporting the findings of this study are available within the article.

The views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated agency of the author.